Let $ABC$ be an acute-angled triangle and $D$ the midpoint of $BC$. Let $E$ be a point on segment $AD$ and $M$ its projection on $BC$. If $N$ and $P$ are the projections of $M$ on $AB$ and $AC$ then the interior angule bisectors of $\angle NMP$ and $\angle NEP$ are parallel.

Given a positive integer $n \geq 2$. Solve the following system of equations: $ \begin{cases} \ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| \\ \ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| \\ &\dots \\ \ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. \\ \end{cases} $

Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$. Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$

Oscar is drawing diagrams with trash can lids and sticks. He draws a triangle $ABC$ and a point $D$ such that $DB$ and $DC$ are tangent to the circumcircle of $ABC$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. If $O$ is the circumcenter of $DB'C'$, help Oscar prove that $AO$ is perpendicular to $BC$. [i]James Lin[/i]

Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.

Let’s call a positive integer [i]interesting[/i] if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting?

The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$, a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.

Alicia earns $ \$20$ per hour, of which $ 1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? $ \textbf{(A)}\ 0.0029 \qquad \textbf{(B)}\ 0.029 \qquad \textbf{(C)}\ 0.29 \qquad \textbf{(D)}\ 2.9 \qquad \textbf{(E)}\ 29$

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

Let $ABCD$ be a parallelogram. Suppose that $E$ is on line $DC$ such that $C$ lies on segment $ED$. Then say lines $AE$ and $BD$ intersect at $X$ and lines $CX$ intersects AB at F. If $AB = 7$,$ BC = 13$, and $CE = 91$, then find $\frac{AF}{FB}$.

Let $t_1,t_2,\ldots,t_k$ be different straight lines in space, where $k>1$. Prove that points $P_i$ on $t_i$, $i=1,\ldots,k$, exist such that $P_{i+1}$ is the projection of $P_i$ on $t_{i+1}$ for $i=1,\ldots,k-1$, and $P_1$ is the projection of $P_k$ on $t_1$.

Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.

In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on $k$.

Let circle $O$ have radius $5$ with diameter $\overline{AE}$. Point $F$ is outside circle $O$ such that lines $\overline{F A}$ and $\overline{F E}$ intersect circle $O$ at points $B$ and $D$, respectively. If $F A = 10$ and $m \angle F AE = 30^o$, then the perimeter of quadrilateral ABDE can be expressed as $a + b\sqrt2 + c\sqrt3 + d\sqrt6$, where $a, b, c$, and $d$ are rational. Find $a + b + c + d$.

Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.

Let $ABC$ be an equilateral triangle whose sides have length $1$. The midpoints of $AB,BC$ are $M,N$ respectively. Points $K,L$ were chosen on $AC$ so that $KLMN$ is a rectangle. Inside this rectangle are three semi-circles with the same radius, as in the picture (the endpoints are on the edges of the rectangle, and the arcs are tangent). Find the minimum possible value of the radii of the semi-circles.

In space are given $24$ points, no three of which are collinear. Suppose that there are exactly $2002$ planes determined by three of these points. Prove that there is a plane containing at least six points.

Consider a triangle $ABC$ and a point $P$ in its interior. Lines $PA$, $PB$, $PC$ intersect $BC$, $CA$, $AB$ at $A', B', C'$ , respectively. Prove that $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$ if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.

Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\ldots \ 2p\}$ are there, the sum of whose elements is divisible by $p$?

Medians $AD, BE$, and $CF$ of triangle $ABC$ intersect at point $M$. Is it possible that the circles with radii $MD, ME$, and $MF$ a) all have areas smaller than the area of triangle $ABC$, b) all have areas greater than the area of triangle $ABC$, c) all have areas equal to the area of triangle $ABC$?

Let $a_1,a_2,\cdots,a_{100}\geq 0$ such that $\max\{a_{i-1}+a_i,a_i+a_{i+1}\}\geq i $ for any $2\leq i\leq 99.$ Find the minimum of $a_1+a_2+\cdots+a_{100}.$

Lunasa, Merlin, and Lyrica each has an instrument. We know the following about the prices of their instruments: (a) If we raise the price of Lunasa's violin by $50\%$ and decrease the price of Merlin's trumpet by $50\%$, the violin will be $\$50$ more expensive than the trumpet; (b) If we raise the price of Merlin's trumpet by $50\%$ and decrease the price of Lyrica's piano by $50\%$, the trumpet will be $\$50$ more expensive than the piano. Given these conditions only, there exist integers $m$ and $n$ such that if we raise the price of Lunasa's violin by $m\%$ and decrease the price of Lyrica's piano by $m\%$, the violin must be exactly $\$n$ more expensive than the piano. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

Let $ f(x)$ be a real function such that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\] and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\] [i]P. Erdos[/i]

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that $$f(x^2 + y^2) = g(xy)$$ holds for all $x, y \in \mathbb{R}^{+}$.

Let $ p(x)\equal{}a_0 \plus{}a_1 x\plus{}...\plus{}a_n x^n$ be a polynomial with nonnegative real coefficients. Suppose that $ p(4)\equal{}2$ and $ p(16)\equal{}8$. Prove that $ p(8) \le 4$ and find all such $ p$ with $ p(8)\equal{}4$.